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Calculus Archive: Questions from August 02, 2022

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  Find \( \operatorname{div} \mathbf{F} \) and curl \( \mathbf{F} \) if \( \mathbf{F}(x, y, z)=4 e^{x y} \mathbf{i}-2 \cos y \mathbf{j}+2 \sin ^{2} z \mathbf{k} \) \( \operatorname{div} \mathbf{F}= \) \
  1 answer

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  Find \( \nabla \times(\nabla \times \mathbf{F}) \) if \( \mathbf{F}(x, y, z)=6 x y \mathbf{j}+5 x y z \mathbf{k} \) \[ \nabla \times(\nabla \times \mathbf{F})=6+5 y i+5 x j \]
  3 answers

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  (11) The solution of \( y^{\prime}-8 y^{\prime}+16 y=0 \) A) \( y=A e^{3 x}+B x e^{3 x} \) B) \( y=A e^{4 x}+B e^{-4 x} \) C) \( y=A e^{4 x}+B x e^{4 x} \) D) \( y=A e^{4 x}+B e^{-2 x} \) (12) The sol
  3 answers
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  Find the domain and range of the function. \[ f(x, y)=\frac{\sqrt{x}}{y} \] Domain: \( \{(x, y): x \) is any real number, \( y \) is any real number \( \} \) \[ \begin{array}{l} \{(x, y): x \geq 0, y
  1 answer

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  Evaluate \( \int_{39}^{61} \int_{0}^{30} \int_{y}^{30} \) \( \frac{x \cos z}{z} d z d y d x \). and the sphere
  1 answer
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  \[ \frac{3}{2} x^{4}+\pi x^{3}+9 x+-4 x^{6}+\sqrt{x}+x^{\pi}-\sin (x)=y \] \( y=\left(5 x^{2}+8 x-5\right)^{6} \), then \( d y / d x \) is:
  3 answers
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  Evaluate the double integral \[ \int_{0}^{2} \int_{1}^{3} x^{2} y d y d x \] \( 16 / 3 \) 16 \( 64 / 3 \) \( 32 / 3 \) 32
  3 answers
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  \[ \int \cos (5 x) \cos (3 x) d x= \] A) \( \frac{\sin (8 x)}{8}+\frac{\sin (2 x)}{2}+c \) B) \( \frac{\sin (8 x)}{16}+\frac{\sin (2 x)}{4}+c \) C) \( \frac{\cos (5 x)}{8}+\frac{\cos (2 x)}{2}+c \) D)
  3 answers

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  \( \frac{\sin ^{3} x+\cos ^{3} x}{1-2 \cos ^{2} x}=\frac{\sec x-\sin x}{\tan x-1} \)
  1 answer
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  Evaluate the integral \[ \int_{(-2,0)}^{(9, \pi / 2)} e^{x} \sin \] A) \( e^{-2} \) B) \( e^{9} \cdot e^{-2} \) C) 0 D) \( e^{9} \)
  3 answers

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  Find the partial derivatives of the function \[ f(x, y)=x y e^{4 y} \] \[ \begin{array}{l} f_{x}(x, y)= \\ f_{y}(x, y)= \\ f_{x y}(x, y)= \\ f_{y x}(x, y)= \end{array} \]
  1 answer
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  Solve for \( \mathrm{y} \). \( y^{\prime}=9 x+x y ; y=-2 \) when \( x=0 \) \[ y= \]
  1 answer

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  Find \( f \) so that \( F=\nabla \mathrm{f} \). If the fuction is not conservative state so. \( F(x, y, z)=\left(e^{y^{2}}-\sin x\right) i+2 x y e^{y^{2}} \mathbf{j}+k \) A) \( f(x, y, z)=\cos x+x z e
  3 answers

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  Evaluate the integral. \[ \int_{(-2,0)}^{(9, \pi / 2)} e^{x} \sin y d x+e^{x} \cos y d y \]
  1 answer
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  Find the derivative of the function. \[ y=\frac{x^{2}+8 x+3}{\sqrt{x}} \] A. \( y^{\prime}=\frac{3 x^{2}+8 x-3}{2 x^{3 / 2}} \) B. \( y^{\prime}=\frac{2 x+8}{x} \) C. \( y^{\prime}=\frac{2 x+8}{2 x^{3
  1 answer

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  \( f(x, y)=\frac{4 x}{y}-\frac{y}{4 x} \)
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  1. For \( f(x, y)=3^{x}+\sin y \) find \( f(0,-2), f(-2,1) \), and \( f(2,1) \)
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  Given \( f(x, y)=-3 x^{4}+6 x^{2} y^{2}+5 y^{6} \) \[ f_{x}(x, y)= \] \( f_{y}(x, y)= \) \( f_{x x}(x, y)= \) \( f_{x y}(x, y)= \) \( f_{y z}(x, y)= \) \( f_{y y}(x, y)= \)
  3 answers
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  Given \( f(x, y)=3 x^{3}+6 x y^{6}-3 y^{5} \), \( f_{x x}(x, y)= \) \[ f_{x y}(x, y)= \] \[ f_{y y}(x, y)= \] \[ f_{y z}(x, y)= \]
  1 answer
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  Given \( f(x, y)=5 x^{6} \ln \left(y^{3}\right) \) \[ f_{x x}(x, y)= \] \[ f_{y y}(x, y)= \]
  1 answer
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  Find the following partial derivatives of \( f(x, y)=x y e^{5 y} \). \[ f_{x}(x, y)= \] \[ f_{y}(x, y)= \] \[ f_{x y}(x, y)= \] \[ f_{y z}(x, y)= \]
  1 answer
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  Compute the second-order partial derivatives of \[ \begin{array}{l} f(x, y)=e^{6 x^{2}+2 y^{2}} \\ \frac{\partial^{2} f}{\partial x^{2}}(x, y)= \\ \frac{\partial^{2} f}{\partial x \partial y}(x, y)= \
  1 answer

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  Find \( \operatorname{div} \mathbf{F} \) and curl \( \mathbf{F} \) if \( \mathbf{F}(x, y, z)=4 e^{x y} \mathbf{i}-4 \cos y \mathbf{j}+\sin ^{2} z \mathbf{k} \) \( \operatorname{div} \mathbf{F}= \) \(
  3 answers

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  ind \( \nabla \cdot(\nabla \times \mathbf{F}) \) if \( \mathbf{F}(x, y, z)=9 \sin x \mathbf{i}+\cos (4 x-5 y) \mathbf{j}+z \mathbf{k} \) \[ \nabla \cdot(\nabla \times \mathbf{F})= \]
  1 answer
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  Find \( \nabla \cdot(\mathbf{F} \times \mathbf{G}) \), if \( \mathbf{F}(x, y, z)=3 x \mathbf{i}+\mathbf{j}+11 y \mathbf{k} \) and \( \mathbf{G}(x, y, z)=x \mathbf{i}+y \mathbf{j}-z \mathbf{k} \). \[ \
  3 answers